Study of Motion of Conductor in a Magnetic Field in the Light of Induced Current

The phenomenon of electromagnetic induction, discovered by Michael Faraday, describes the process by which a current is induced in a conductor when it is exposed to a changing magnetic field. This principle is fundamental to the operation of generators, transformers, and many types of electrical machinery. In this article, we will explore the motion of a conductor in a magnetic field and how it leads to the induction of an electric current.

Faraday's Law of Electromagnetic Induction

Faraday's law states that the induced electromotive force (EMF) in any closed circuit is equal to the negative of the time rate of change of the magnetic flux through the circuit. Mathematically, it is expressed as:

$$ \mathcal{E} = -\frac{d\Phi_B}{dt} $$

where $\mathcal{E}$ is the induced EMF and $\Phi_B$ is the magnetic flux.

Lenz's Law

Lenz's law gives the direction of the induced current. It states that the direction of the induced current is such that it opposes the change in magnetic flux that produced it. This is a manifestation of the conservation of energy.

Motion of a Conductor in a Magnetic Field

When a conductor moves through a magnetic field, or when the magnetic field around a conductor changes, an EMF is induced across the conductor. If the conductor is part of a closed circuit, this EMF will cause a current to flow.

Important Points:

• The magnitude of the induced EMF is proportional to the rate of change of magnetic flux.
• The direction of the induced current is given by Lenz's law.
• The induced current will continue to flow as long as there is relative motion between the conductor and the magnetic field.

Formulas

The induced EMF can also be calculated using the formula:

$$ \mathcal{E} = B \cdot l \cdot v \cdot \sin(\theta) $$

where:

• $B$ is the magnetic field strength
• $l$ is the length of the conductor
• $v$ is the velocity of the conductor
• $\theta$ is the angle between the direction of motion and the magnetic field

Examples

Example 1: Straight Conductor Moving in a Magnetic Field

Consider a straight conductor of length $l$ moving with velocity $v$ perpendicular to a uniform magnetic field $B$. The induced EMF across the conductor is given by:

$$ \mathcal{E} = B \cdot l \cdot v $$

since $\sin(90^\circ) = 1$.

Example 2: Rotating Loop in a Magnetic Field

A rectangular loop of wire with sides $l$ and $w$ rotates with an angular velocity $\omega$ in a uniform magnetic field $B$. The magnetic flux through the loop changes with time, and the induced EMF can be calculated using Faraday's law.

Differences and Important Points

Aspect	Motion of Conductor in Magnetic Field	Stationary Conductor in Changing Magnetic Field
Cause of Induced EMF	Relative motion	Time-varying magnetic field
Calculation of Induced EMF	$\mathcal{E} = B \cdot l \cdot v \cdot \sin(\theta)$	$\mathcal{E} = -\frac{d\Phi_B}{dt}$
Direction of Induced Current	Given by Lenz's Law	Given by Lenz's Law
Requirement for Induction	Conductor must cut magnetic field lines	Magnetic field lines through the conductor must change

Conclusion

The study of the motion of a conductor in a magnetic field reveals the principles of electromagnetic induction, which is the basis for generating electric power and the operation of many electrical devices. Understanding these principles is crucial for the design and analysis of electrical machinery and systems.